The demand for increasingly small and lightweight products require micro-scale components made of materials which are durable and light. Polymers have therefore become a popular choice since they can be used to produce materials which meet industrial requirements. Many of these polymers are viscoelastic fluids. The reduction in the sizes of components make physical experimentation difficult and costly. Therefore computational tools are being sought to replace old methods of testing. This research has been concerned with the development of a finite volume algorithm for viscoelastic flow which can be readily applied to real world applications. A major part of the research involved the implementation of the Oldroyd-B constitutive equations and associated solution methods, in the 3-D multi-physics software environment PHYSICA+. This provides an unstructured finite volume solution technique for viscoelastic flow. This algorithm is validated using the 4:1 planar contraction and results are reported. The developed viscoelastic algorithm has also been coupled with two interface tracking techniques one of which includes surface tension effects. These techniques are the Scalar Equation Algorithm (SEA) and the Level Set Method (LSM). With both techniques the algorithms are able to take into account flow effects from both fluids (ie. air and polymer) in a two-fluid system. The LSM technique maintains a sharp interface overcoming the smearing of the interface which generally affects interface tracking techniques on Eulerian fixed grids, for example SEA, and enables the curvature of the interface to be calculated accurately to implement surface tension effects. This integrated viscoelastic flow solver and free surface algorithm is then illustrated by predicting two industrial flow processes as used in the electronic packaging industry.